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Lets say I have a model:

$$ y_{i,t}= \sum_{k \neq -1} \beta_k \times treat_i \times \mathbf{1}_{K = k} + \lambda_t + \mu_i + e_{i,t}, $$

where $k$ indicates event time, and treatment takes place at even time = $0$. The variable $treat_i$ is a dummy for treatment status, and $\mathbf{1}_{K = k}$ is an indicator if event time = $k$. These models are usually used with differences in differences to show pre-trends and trace out dynamic effects. I am wondering how you would interpret a given coefficient $\beta_k$ for say, $k = 2$:

$$ (E[y|k=2,treatment]-E[y|k=2,control])- (E[y|k=-1,treatment]-E[y|k=-1,control])? $$

aka a difference in difference for the event time $k = 2$ compared to event time $-1$?

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  • $\begingroup$ Your interpretation looks about right. If I am correct, this model is fully saturated. It assesses all period effects before and after $k = -1$ (i.e., the omitted baseline period). Correct? $\endgroup$
    – Thomas Bilach
    Commented Aug 7, 2020 at 0:35
  • $\begingroup$ Yes that is correct. So that makes sense, so each coefficient can be interpreted like it is a difference in difference estimate for that event time relative to k= -1? $\endgroup$
    – Steve
    Commented Aug 7, 2020 at 0:46
  • $\begingroup$ What is the difference between mu and lambda? Should one of them have an i subscript? $\endgroup$
    – dimitriy
    Commented Aug 7, 2020 at 5:43
  • $\begingroup$ Yes, I corrected that in the original question. Thanks for pointing that out! $\endgroup$
    – Steve
    Commented Aug 7, 2020 at 5:51

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Your interpretation is sufficient. Your model is fully saturated with time dummies. The omitted time dummy (i.e., $k = -1$) is your reference period. You could use a more distant pre-event period but most papers I have encountered use the period before treatment as the baseline.

I wouldn't get too caught up with interpreting all these interactions. Rather, organize them into a table where you can easily juxtapose your coefficients or estimate more complex dynamic specifications. If you're partial to graphical displays, then try plotting your coefficients to show how effects evolve before and after treatment exposure. A picture is worth a thousand words.

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